# A property for locally convex *-algebras related to property (T) and character amenability

Xiao Chen; Anthony To-Ming Lau; Chi-Keung Ng

Studia Mathematica (2015)

- Volume: 227, Issue: 3, page 259-286
- ISSN: 0039-3223

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topXiao Chen, Anthony To-Ming Lau, and Chi-Keung Ng. "A property for locally convex *-algebras related to property (T) and character amenability." Studia Mathematica 227.3 (2015): 259-286. <http://eudml.org/doc/285929>.

@article{XiaoChen2015,

abstract = {For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let $C_\{c\}(G)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $(C_\{c\}(G),ε_\{G\})$ has property (FH) if and only if G has property (T). On the other hand, many Banach algebras equipped with canonical characters have property (FH) (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property (FH) and character amenablility, we obtain characterizations of property (T), amenability and compactness of G in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These three sets of characterizations can be regarded as analogues of one another. Moreover, we show that G is compact if and only if the normed algebra $\{f ∈ C_\{c\}(G): ∫_\{G\} f(t)dt =0\}$ (under $||·||_\{L¹(G)\}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.},

author = {Xiao Chen, Anthony To-Ming Lau, Chi-Keung Ng},

journal = {Studia Mathematica},

keywords = {locally compact groups; property ; convolution algebras; -algebras; Hochschild cohomology; property ; character amenability; fixed point properties},

language = {eng},

number = {3},

pages = {259-286},

title = {A property for locally convex *-algebras related to property (T) and character amenability},

url = {http://eudml.org/doc/285929},

volume = {227},

year = {2015},

}

TY - JOUR

AU - Xiao Chen

AU - Anthony To-Ming Lau

AU - Chi-Keung Ng

TI - A property for locally convex *-algebras related to property (T) and character amenability

JO - Studia Mathematica

PY - 2015

VL - 227

IS - 3

SP - 259

EP - 286

AB - For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let $C_{c}(G)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $(C_{c}(G),ε_{G})$ has property (FH) if and only if G has property (T). On the other hand, many Banach algebras equipped with canonical characters have property (FH) (e.g., those defined by a nice locally compact quantum group). Furthermore, through our studies on both property (FH) and character amenablility, we obtain characterizations of property (T), amenability and compactness of G in terms of the vanishing of one-sided cohomology of certain topological algebras, as well as in terms of fixed point properties. These three sets of characterizations can be regarded as analogues of one another. Moreover, we show that G is compact if and only if the normed algebra ${f ∈ C_{c}(G): ∫_{G} f(t)dt =0}$ (under $||·||_{L¹(G)}$) admits a bounded approximate identity with the supports of all its elements being contained in a common compact set.

LA - eng

KW - locally compact groups; property ; convolution algebras; -algebras; Hochschild cohomology; property ; character amenability; fixed point properties

UR - http://eudml.org/doc/285929

ER -

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