# Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

Studia Mathematica (1998)

- Volume: 129, Issue: 3, page 207-223
- ISSN: 0039-3223

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topHu, Zhiguo. "Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups." Studia Mathematica 129.3 (1998): 207-223. <http://eudml.org/doc/216501>.

@article{Hu1998,

abstract = {Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_A$. In this paper, we study the norm spectra of elements of $\overline\{span\} Σ_A$ and present some applications. In particular, we characterize the discreteness of $Σ_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $φ ∈ \overline\{\} Σ_A \ \{0\}$, φ has a nonempty norm spectrum. For a locally compact group G, let $ℳ_2^\{d\}(Ĝ)$ denote the C*-algebra generated by left translation operators on $L^2(G)$ and $G_\{d\}$ denote the discrete group G. We prove that the Fourier algebra $A(G)$ has property (S) iff the canonical trace on $ℳ_2^\{d\}(Ĝ)$ is faithful iff $ℳ_2^\{d\} (Ĝ)≅ ℳ_2^\{d\} (Ĝ_\{d\})$. This provides an answer to the isomorphism problem of the two C*-algebras and generalizes the so-called “uniqueness theorem” on the group algebra $L^1(G)$ of a locally compact abelian group G. We also prove that $G_\{d\}$ is amenable iff G is amenable and the Figà-Talamanca-Herz algebra $A_p(G)$ has property (S) for all p.},

author = {Hu, Zhiguo},

journal = {Studia Mathematica},

keywords = {spectrum; synthesizable ideal; locally compact group; Fourier algebra; Figà-Talamanca-Herz algebra; amenability; -algebra; semisimple commutative regular Tauberian Banach algebra; norm spectrum; Figá-Talamanca-Herz algebra; amenable group},

language = {eng},

number = {3},

pages = {207-223},

title = {Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups},

url = {http://eudml.org/doc/216501},

volume = {129},

year = {1998},

}

TY - JOUR

AU - Hu, Zhiguo

TI - Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

JO - Studia Mathematica

PY - 1998

VL - 129

IS - 3

SP - 207

EP - 223

AB - Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_A$. In this paper, we study the norm spectra of elements of $\overline{span} Σ_A$ and present some applications. In particular, we characterize the discreteness of $Σ_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $φ ∈ \overline{} Σ_A \ {0}$, φ has a nonempty norm spectrum. For a locally compact group G, let $ℳ_2^{d}(Ĝ)$ denote the C*-algebra generated by left translation operators on $L^2(G)$ and $G_{d}$ denote the discrete group G. We prove that the Fourier algebra $A(G)$ has property (S) iff the canonical trace on $ℳ_2^{d}(Ĝ)$ is faithful iff $ℳ_2^{d} (Ĝ)≅ ℳ_2^{d} (Ĝ_{d})$. This provides an answer to the isomorphism problem of the two C*-algebras and generalizes the so-called “uniqueness theorem” on the group algebra $L^1(G)$ of a locally compact abelian group G. We also prove that $G_{d}$ is amenable iff G is amenable and the Figà-Talamanca-Herz algebra $A_p(G)$ has property (S) for all p.

LA - eng

KW - spectrum; synthesizable ideal; locally compact group; Fourier algebra; Figà-Talamanca-Herz algebra; amenability; -algebra; semisimple commutative regular Tauberian Banach algebra; norm spectrum; Figá-Talamanca-Herz algebra; amenable group

UR - http://eudml.org/doc/216501

ER -

## References

top- [1] E. Bédos, On the C*-algebra generated by the left translation of a locally compact group, Proc. Amer. Math. Soc. 120 (1994), 603-608. Zbl0806.22004
- [2] M. Bekka, E. Kaniuth, A. T. Lau and G. Schlichting, On C*-algebras associated with locally compact groups, ibid. 124 (1996), 3151-3158. Zbl0861.43002
- [3] M. Bekka, A. T. Lau and G. Schlichting, On invariant subalgebras of the Fourier-Stieljes algebra of a locally compact group, Math. Ann. 294 (1992), 513-522. Zbl0787.22005
- [4] M. Bekka and A. Valette, On duals of Lie groups made discrete, J. Reine Angew. Math. 439 (1993), 1-10.
- [5] J. J. Benedetto, Spectral Synthesis, Academic Press, New York, 1975. Zbl0314.43011
- [6] C. Chou, Almost periodic operators in VN(G), Trans. Amer. Math. Soc. 317 (1990), 229-253. Zbl0694.43007
- [7] C. Chou, A. T. Lau and J. Rosenblatt, Approximation of compact operators by sums of translations, Illinois J. Math. 29 (1985), 340-350. Zbl0546.22007
- [8] M. G. Cowling and J. J. F. Fournier, Inclusions and noninclusions of spaces of convolution operators, Trans. Amer. Math. Soc. 221 (1976), 59-95. Zbl0331.43007
- [9] C. De Vito, Characterizations of those ideals in ${L}_{1}\left(\mathbb{R}\right)$ which can be synthesized, Math. Ann. 203 (1973), 171-173.
- [10] V. G. Drinfel'd, Finitely additive measures on S² and S³, invariant with respect to rotations, Functional Anal. Appl. 18 (1984), 245-246.
- [11] J. Duncan and S. A. R. Husseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sec. A 84 (1979), 309-325. Zbl0427.46028
- [12] C. F. Dunkl and D. E. Ramirez, C*-algebras generated by Fourier-Stieltjes transformations, Trans. Amer. Math. Soc. 164 (1972), 435-441. Zbl0211.15903
- [13] C. F. Dunkl and D. E. Ramirez, Weakly almost periodic functionals on the Fourier algebra, ibid. 185 (1973), 501-514. Zbl0271.43009
- [14] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. Zbl0169.46403
- [15] B. Forrest, Arens regularity and discrete groups, Pacific J. Math. 151 (1991), 217-227. Zbl0746.43002
- [16] E. E. Granirer, A characterisation of discreteness for locally compact groups in terms of the Banach algebras ${A}_{p}\left(G\right)$, Proc. Amer. Math. Soc. 54 (1976), 189-192. Zbl0317.43010
- [17] E. E. Granirer, On some spaces of linear functionals on the algebras ${A}_{p}\left(G\right)$ for locally compact groups, Colloq. Math. 52 (1987), 119-132. Zbl0649.43004
- [18] E. E. Granirer, On convolution operators which are far from being convolution by a bounded measure. Expository memoir, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 187-204; corrigendum: ibid. 14 (1992), 118. Zbl0791.43003
- [19] E. E. Granirer, On convolution operators with small support which are far from being convolution by a bounded measure, Colloq. Math. 67 (1994), 33-60; erratum: ibid. 69 (1995), 155. Zbl0841.43008
- [20] E. E. Granirer, On the set of topologically invariant means on an algebra of convolution operators on ${L}^{p}\left(G\right)$, Proc. Amer. Math. Soc. 124 (1996), 3399-3406. Zbl0853.43007
- [21] F. Greenleaf, Invariant Means of Topological Groups and Their Applications, Van Nostrand Math. Stud. 16, Van Nostrand, New York, 1969. Zbl0174.19001
- [22] C. Herz, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82. Zbl0216.15606
- [23] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (3) (1973), 91-123. Zbl0257.43007
- [24] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vols. I, II, Springer, New York, 1970. Zbl0213.40103
- [25] Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publ., New York, 1976.
- [26] A. T. Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39-59. Zbl0436.43007
- [27] A. T. Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, ibid. 267 (1981), 53-63. Zbl0489.43006
- [28] A. T. Lau and V. Losert, The C*-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), 1-30. Zbl0788.22006
- [29] A. L. T. Paterson, Amenability, Amer. Math. Soc., Providence, R.I., 1988.
- [30] J. P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. Zbl0597.43001
- [31] P. F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285-291. Zbl0226.46065
- [32] A. Ülger, Some results about the spectrum of commutative Banach algebras under the weak topology and applications, Monatsh. Math. 121 (1996), 353-379. Zbl0851.46036
- [33] G. Zeller-Meier, Représentations fidèles des produits croisés, C. R. Acad. Sci. Paris Sér. A 264 (1967), 679-682. Zbl0152.13302

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