# A very proper Heisenberg–Lie Banach *-algebra

A very proper Heisenberg–Lie Banach *-algebra For each pair of non-zero real numbers q 1 and q 2, Laustsen and Silvestrov have constructed a unital Banach *-algebra $${\fancyscript{C}_{q_1,q_2}}$$ which contains a universal normalized solution to the *-algebraic (q 1, q 2)-deformed Heisenberg–Lie commutation relations. We show that for (q 1, q 2) = (−1, 1), this Banach *-algebra is very proper; that is, if $${M\in\mathbb{N}}$$ and $${a_1, \ldots, a_M}$$ are elements of $${\fancyscript{C}_{-1,1}}$$ such that $${\sum_{m=1}^M a_m^*a_m=0}$$ , then necessarily $${a_1=a_2=\cdots=a_M=0}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A very proper Heisenberg–Lie Banach *-algebra

, Volume 16 (1) – Feb 12, 2011
13 pages

/lp/springer_journal/a-very-proper-heisenberg-lie-banach-algebra-UUMwLq2Y0x
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0111-2
Publisher site
See Article on Publisher Site

### Abstract

For each pair of non-zero real numbers q 1 and q 2, Laustsen and Silvestrov have constructed a unital Banach *-algebra $${\fancyscript{C}_{q_1,q_2}}$$ which contains a universal normalized solution to the *-algebraic (q 1, q 2)-deformed Heisenberg–Lie commutation relations. We show that for (q 1, q 2) = (−1, 1), this Banach *-algebra is very proper; that is, if $${M\in\mathbb{N}}$$ and $${a_1, \ldots, a_M}$$ are elements of $${\fancyscript{C}_{-1,1}}$$ such that $${\sum_{m=1}^M a_m^*a_m=0}$$ , then necessarily $${a_1=a_2=\cdots=a_M=0}$$ .

### Journal

PositivitySpringer Journals

Published: Feb 12, 2011

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