TY - GEN
T1 - MR3730338 Reviewed de Jeu, Marcel(NL-LEID-MI); Tomiyama, Jun(J-TOKYM)The closure of ideals of ℓ1(Σ) in its enveloping C∗-algebra. (English summary) Adv. Oper. Theory 3 (2018), no. 1, 42–52. 46K10 (47L65 54H20)
AU - Tschinke, Francesco
PY - 2018
Y1 - 2018
N2 - Given a compact Hausdorff space X and a homeomorphism σ on X, denote by Σ=(X,σ) a topological dynamical system. Then the associated Banach ∗-algebra ℓ1(Σ) is defined asℓ1(Σ)={a:Z→C(X), ∥a∥:=∑n∈Z∥a(n)∥<∞}with a crossed product–type product(aa′)(n)=∑k∈Za(k)⋅αk(a′(n−k))and involutiona∗(n)=αn(a(−n))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯,where C(X) denote the space of complex-valued continuous functions on X, and α(f):=f∘σ−1 for f∈C(X). If C∗(Σ) is the enveloping C∗-algebra of ℓ1(Σ), considering a primitive ideal I of ℓ1(Σ), the authors show that there exists a ∗-representation π of ℓ1(Σ) on Hilbert space such that the kernel is I, and that the closure in C∗(Σ) of an ideal of ℓ1(Σ) is an ideal of C∗(Σ).
AB - Given a compact Hausdorff space X and a homeomorphism σ on X, denote by Σ=(X,σ) a topological dynamical system. Then the associated Banach ∗-algebra ℓ1(Σ) is defined asℓ1(Σ)={a:Z→C(X), ∥a∥:=∑n∈Z∥a(n)∥<∞}with a crossed product–type product(aa′)(n)=∑k∈Za(k)⋅αk(a′(n−k))and involutiona∗(n)=αn(a(−n))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯,where C(X) denote the space of complex-valued continuous functions on X, and α(f):=f∘σ−1 for f∈C(X). If C∗(Σ) is the enveloping C∗-algebra of ℓ1(Σ), considering a primitive ideal I of ℓ1(Σ), the authors show that there exists a ∗-representation π of ℓ1(Σ) on Hilbert space such that the kernel is I, and that the closure in C∗(Σ) of an ideal of ℓ1(Σ) is an ideal of C∗(Σ).
UR - http://hdl.handle.net/10447/325776
UR - https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=RVCN&pg5=TI&pg6=RVCN&pg7=ALLF&pg8=ET&review_format=html&s4=tschinke&s5=&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=3730338
M3 - Other contribution
ER -